Time: 3 hours                                                                             Maximum marks: 80

General Instructions:

1. This Question paper contains - five sections A, B, C, D and E. Each section is compulsory. However, there are internal choices in some questions.

2. Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark each.

3. Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.

4. Section C has 6 Short Answer (SA)-type questions of 3 marks each.

5. Section D has 4 Long Answer (LA)-type questions of 5 marks each.

6. Section E has 3 source based/case based/passage based/integrated units of assessment of 4 marks each with sub-parts.

 

Section –A

(Multiple Choice Questions)

Question 1:

If for a square matrix A, A² - 3A + I = O and A^-1 = xA + yI, then the value of x + y is

(a) 2

(b) 3

(c) 4

(d) 5

 

Question 2:

The value of ʃ [x² * (1 – )] dx

(a)  + x + C

(b)  – x + C

(c) x³ –  + C

(d) x³ +  + C

 

Question 3:

If 2i + j – k and i – 4j + λk are perpendicular to each other, then λ is equal to  

(a) -1

(b) -2

(c) -3

(d) -4

 

Question 4:

If a + b = i and a = 2i – 2j + 2k then the value of |b| is

(a) 3

(b)

(c)

(d)

 

Question 5:

The direction ratios of a line parallel to y-axis are

(a) 0, 0, 1

(b) 1, 1, 0

(c) 1, 0, 1

(d) 0, 1, 0

 

Question 6:

If P(B) = 0.5 and P (A ∩ B) = 0.32, then the value of P( ) is

(a) 0, 0, 1

(b) 1, 1, 0

(c) 1, 0, 1

(d) 0, 1, 0

 

Question 7:

The order and degree of differential equation  + sec( ) = 0

(a) 2, 1

(b) 2, not defined

(c) not defined, not defined

(d) nor defined, 1

 

Question 8:

ABCD is a rhombus whose diagonals intersect at E. Then EA + EB + EC + ED equals to

(a) ED

(b) 2ED

(c) 3ED

(d) 0

 

Question 9:

The value of ꭍ₄⁵ e^x dx is

(a) e(e – 1)

(b) e²(e – 1)

(c) e³(e – 1)

(d) e⁴(e – 1)

 

Question 10:

A relation R in the set N of natural numbers defined as R = {(x, y): y = x + 5 and x < 4} then it is

(a) reflexive, symmetric and transitive

(b) reflexive, symmetric but not transitive

(c) either reflexive, or symmetric, or transitive

(d) neither reflexive, nor symmetric, nor transitive

 

Question 11:

If A =    2       -1      

             3        5    , then the value of |2A| is

(a) 14

(b) 20

(c) 28

(d) 43

 

Question 12:

The value of A so that the function defined by

f(x) =    A(x² – 2x),    if x ≤ 0

             4x + 1,          if x > 0

is continuous at x = 1, is

(a) 4

(b) -2

(c)

(d) For any value of A

 

Question 13:

The point which does not lie in the half-plane 2x + 3y - 12 < 0 is:

(a) (2, 1)

(b) (1, 2)

(c) (-2, 3)

(d) (2, 3)

 

Question 14:

If a = 4i + 6j and b = 3j + 4k then the projection of a along b is

(a) (-3j – 4k)

(b) (-3j + 4k)

(c) (3j – 4k)

(d) (3j + 4k)

 

Question 15:

If A is a square matrix of order 3 and |A| = 5, then the value of |2A′| is

(a) -10

(b) 10

(c) -40

(d) 40

 

Question 16:

The solution of the differential equation dy = (1 + y²) dx is

(a) y = tan x + C

(b) y = tan (x + C)

(c) x = tan y + C

(d) x = tan (y + C)

 

Question 17:

Suppose a number x is chosen from the numbers -2, -1, 0, 1, 2. What will be the probability of x² > 0?

(a)

(b)

(c)

(d)

 

Question 18:

The points which lie outside the feasible region in the given figure are

 

(a) (60, 0), (10, 50)

(b) (60, 0), (0, 100)

(c) (10, 50), (20, 0)

(d) (60, 0), (0, 60)

 

ASSERTION-REASON BASED QUESTIONS

In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R).

Choose the correct answer out of the following choices.

(a) Both (A) and (R) are true and (R) is the correct explanation of (A).

(b) Both (A) and (R) are true but (R) is not the correct explanation of (A).

(c) (A) is true but (R) is false.

(d) (A) is false but (R) is true.

 

Question 19:

ASSERTION (A): If R is the relation defined in the set A = {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1}, then R is reflexive.

REASON (R): The relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric.

 

Question 20:

ASSERTION (A): f(x) is continuous at x = a iff  exists and equals to f(a).

REASON (R): If f(x) is continuous at a point then  is also continuous at that point.

 

Section – B

Question 21:

Find the value of sec^-1[sec(-30°)]

 

Question 22:

Find the interval in which the function f(x) = log sin x is strictly increasing.

 

Question 23:

Find the slope of the tangent to the curve y = 10x⁴ − 2x + 5 at x = 2.

 

Question 24:

Evaluate:  dx

 

Question 25:

Find the equation of the normal to curve y² = 2x + 5 at the point (2, 3).

 

Section – C

Question 26:

Find the value of  dx

 

Question 27:

The random variable X has probability distribution P(X) of the following form, where k is some number:

              K,            if x = 0

P(X) =   2k,           if x = 1

              3k,          if x = 2 

               0,           otherwise

(a) Determine the value of k.                          (b) Find P(X ≥ 2)

 

Question 28:

Evaluate:  dx

 

Question 29:

Find the general solution of  = 2x *

 

Question 30:

Maximize Z = 5x + 3y

subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0

 

Question 31:

If the solution of sin³ x *  = sin y is A * cos y + B * cos x + C * cos 3x then prove that A + B + C =

 

Section – D

Question 32:

Make a rough sketch of the region of the curves y = x² + 2, y = x, x = 0 and x = 3 and find the area of the region using the method of integration.

 

Question 33:

Show that the function f: R -> [ , ] defined as f(x) = , is surjective but not injective.

 

Question 34:

Using the matrix method, solve the following system of linear equations

 

 

 

 

Question 35:

The two adjacent sides of a parallelogram are 2i – 4j + 5k and i – 2j – 3k. Find the unit vector parallel to its diagonal. Also, find its area.

 

Section – E

Question 36:

Rihana was working on a school survey project where she was calculating the average number of hours spend on study for different students selected at random. At the end of her survey, she prepared the following report related to data.

Let X denotes the average number of hours spent on study by students. The probability that X can take the values x, has the following form where k is a constant.

                     0.4,          if x = 0

 P(X = x) =   kx,            if x = 1 or 2

                     k(4 - x)    if x = 3 or 4   

                     0,             otherwise 

Based on the above information, answer the following questions.

(i) What will be the value of k?

(ii) Find the probability that the average study time of students more than 1 hour?

(iii) Find the probability that the average study time of students exactly 2 hours?

 

Question 37:

A rumour on facebook spreads in a population of 10000 people at rate proportional to the product of the number of people who have heard it and the number of people who have not heard. Also, it is given that 500 people initiate the rumour and a total of 2000 people know the rumour after 3 hours.

Based on the above information given, answer the following questions.

(i) If y(t) denotes the number of people who know the rumour at an instant t, then the maximum value of y(t) is

(a) 100                       (b) 500                      (c) 2000                        (d) 10000

(ii)  is proportional to

(a)                            (b) ky(10000 - y)     (c) k(10000 - y)            (d) None of these

(iii) The value of y(3) is

(a) 100                      (b) 200                      (c) 500                          (d) 2000

 

Question 38:

Rajeev rides his car at 25 . He has to spend Rs 2 per km on diesel and if he rides it at a faster speed of 40 , the diesel cost increases to Rs 5 per km. He has Rs 100 to spend on the diesel. Let he travels x kms with speed of 25  and y kms with speed of 40 . The feasible region for the LPP has shown below.

Based on the above information given, answer the following questions.

(i) The point of intersection of lines l₁ and l₂ is

(a) ( )                 (b) ( )                  (c) ( )                       (d) ( )

(ii) If Z = x + y be the objective function and max Z = 30 then the maximum value occurs at the point

(a) ( )                 (b) (0, 0)                      (c) (25, 0                          (d) (0, 20)

(iii) If Z = 6x - 9y be the objective function then the maximum value of Z is

(a) 120                      (b) 130                      (c) 0                          (d) 150

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